Biomedical Engineering Reference
In-Depth Information
1
s
A
(
r
1
) α (
s
1
)
1
s
A
(
r
1
) β (
s
1
)
1
s
B
(
r
1
) α (
s
1
)
1
s
B
(
r
1
) β (
s
1
)
D
5
=
D
6
=
1
s
A
(
r
2
) α (
s
2
)
1
s
A
(
r
2
) β (
s
2
)
1
s
B
(
r
2
) α (
s
2
)
1
s
B
(
r
2
) β (
s
2
)
It is useful to combine the
D
's into spin eigenfunctions, since the spin operators commute
with the molecular Hamiltonian. The advantage is that we only need take combinations
of those wavefunctions having the same spinquantum numbers when seeking to improve
our description of the electronic states (Table 15.2). The combination Ψ
1
=
D
2
is a
singlet spin state and is said to represent the covalent bond, since it gives an equal sharing
to the two equivalent 1s orbitals by the two electrons. Ψ
2
through Ψ
4
correspond to the
first excited state, which is a triplet spin state. They have the same energy in the absence
of an external magnetic field. Ψ
5
and Ψ
6
are called ionic terms, because they represent an
electron density distribution in which both electrons are associated with the same nucleus.
Heitler and London included Ψ
1
(for the electronic ground state) and Ψ
2
through Ψ
4
(for
the excited triplet state) in their original calculation. The necessary integrals needed for a
variational calculation are given in Heitler and London's paper, and in the paper by Sugiura
(1937). We often refer to the Heitler-London approach as the
valence bond
(VB) method
and I will use the two descriptors interchangeably. It was the first successful treatment of
an electron pair bond.
D
1
−
Table 15.2
Dihydrogen elementary valence bond (Heitler-London)
calculation
Ψ
Combination
S
S
Comment
Ψ
1
D
1
−
D
2
0
0
Covalent ground state
Ψ
2
D
1
+
D
2
1
0
Excited triplet state
D
3
1
1
Excited triplet state
Ψ
3
D
4
1
−
1
Excited triplet state
Ψ
4
Ψ
5
D
5
0
0
Ionic term
Ψ
6
D
6
0
0
Ionic term
The energy corresponding to Ψ
1
is sometimes written
J
+
K
ε
=
1
+
S
2
1
s
A
(
r
1
) 1
s
B
(
r
2
)
H
1
s
A
(
r
1
) 1
s
B
(
r
2
) dτ
1
dτ
2
J
=
(15.9)
1
s
A
(
r
1
) 1
s
B
(
r
2
)
H
1
s
A
(
r
2
) 1
s
B
(
r
1
) dτ
1
dτ
2
K
=
S
=
1
s
A
(
r
1
)1
s
B
(
r
1
) dτ
and early 'explanations' of chemical bonding focused on the
Coulomb
(
=
J
/1
+
S
2
) and
exchange
(
S
2
) contributions to molecular energies. Several improvements were
made to the simple VB treatment of dihydrogen, for example treating the orbital exponent
as a variational parameter, and inclusion of the ionic terms once again correctly weighted
by use of the variation principle. This latter procedure is referred to as
configuration
interaction
(CI).
=
K
/1
+
